Simplify and expand the following expression: $ \dfrac{y - 5}{3y - 4}-\dfrac{2y - 6}{2y - 5} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(3y - 4)(2y - 5)$ Multiply the first term by $\dfrac{2y - 5}{2y - 5}$ $ \begin{align*} \dfrac{y - 5}{3y - 4} \times \dfrac{2y - 5}{2y - 5} & = \dfrac{(y - 5)(2y - 5)}{(3y - 4)(2y - 5)} \\ & = \dfrac{2y^2 - 15y + 25}{(3y - 4)(2y - 5)}\end{align*} $ Multiply the second term by $\dfrac{3y - 4}{3y - 4}$ $ \begin{align*} \dfrac{2y - 6}{2y - 5} \times \dfrac{3y - 4}{3y - 4} & = \dfrac{(2y - 6)(3y - 4)}{(2y - 5)(3y - 4)} \\ & = \dfrac{6y^2 - 26y + 24}{(2y - 5)(3y - 4)}\end{align*} $ Now we have: $ = \dfrac{2y^2 - 15y + 25}{(3y - 4)(2y - 5)} - \dfrac{6y^2 - 26y + 24}{(2y - 5)(3y - 4)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2y^2 - 15y + 25 - (6y^2 - 26y + 24)}{(3y - 4)(2y - 5)} $ $ = \dfrac{2y^2 - 15y + 25 - 6y^2 + 26y - 24}{(3y - 4)(2y - 5)} $ $ = \dfrac{-4y^2 + 11y + 1}{(3y - 4)(2y - 5)}$ Expand the denominator: $ = \dfrac{-4y^2 + 11y + 1}{6y^2 - 23y + 20}$